Efficient Structured Multifrontal Factorization for General Large Sparse Matrices

Abstract

Rank structures provide an opportunity to develop new efficient numerical methods for practical problems, when the off-diagonal blocks of certain dense intermediate matrices have small (numerical) ranks. In this work, we present a framework of structured direct factorizations for general sparse matrices, including discretized PDEs on general meshes, based on the multifrontal method and hierarchically semiseparable (HSS) matrices. We prove the idea of replacing certain complex structured operations by fast simple ones performed on compact reduced matrix forms. Such forms result from the hierarchical factorization of a tree-structured HSS matrix in a ULV-type scheme, so that the tree structure is reduced into a single node, the root of the original tree. This idea is shown to be very useful in the partial ULV factorization of an HSS matrix (for quickly computing Schur complements) as well as the solution stage. These techniques are then built into the multifrontal method for sparse factorizations after nested dissection, so as to convert the intermediate dense factorizations into fast structured ones. This method keeps certain Schur complements dense so as to avoid complicated data assembly, and is much simpler and more general than some existing methods. In particular, if the matrix arises from the discretization of certain PDEs, the factorization costs roughly O(n) flops in two dimensions, and roughly O(n 4/3) flops or less in three dimensions. The solution cost and memory are nearly O(n) in both cases. These counts are obtained with an idea of rank relaxation, so that this method is more generally applicable to problems where the intermediate off-diagonal ranks are not small. We demonstrate the performance of the method with two-and three-dimensional discretized equations, as well as various examples from a sparse matrix collection. The ideas here are also useful in future developments of fast structured solvers.

A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions

“…The ULV factors associated with each tree node are stored in the corresponding processes. The partial ULV factorization reduces F i;1,1 to a final reduced matrixD k [30].…”

“…The details are given in Algorithm 1. An important observation from this update process is that the HSS rank of U i is bounded by that of F i [30].…”

“…That is, the nested HSS structure enables the Schur complement computation to be conveniently done via some reduced matrices, so that only certain generators of the frontal matrix need to be quickly updated. Such a mathematical scheme was designed and proved in [30,31]. For 3D problems, the HSS structure is indeed less efficient than more advanced structures such as H 2 and multilayer hierarchical structures [32].…”

“…Here, roughly speaking, a separator j is a neighbor of a separator i if j is an ancestor of i and the L factor in the LU factorization of A has nonzero entries in L| s j ×s i . We try to keep nearby vertices close to each other after reordering, so as to preserve the inherent low-rank structures of the intermediate Schur complements [30]. The details are as follows.…”

“…The class of structured multifrontal methods [30,31,33] is based on the idea that, for certain matrices A, the local dense frontal and update matrices are rank structured. For some cases, F i and U i may be approximated by rank structured forms such as HSS forms.…”